Question: A circle with radius $2$ has a sector with a $\dfrac{1}{2}\pi$ radian central angle. What is the area of the sector? ${4\pi}$ $\color{#9D38BD}{\dfrac{1}{2}\pi}$ ${1\pi}$ ${2}$
Explanation: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (2)^2$ $A_c = 4\pi$ The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{1}{2}\pi \div 2 \pi = \dfrac{A_s}{4\pi}$ $\dfrac{1}{4} = \dfrac{A_s}{4\pi}$ $\dfrac{1}{4} \times 4\pi = A_s$ $1\pi = A_s$